You can do surprisingly large amounts of things with old technology. The hard part will be fighting with the enemy.
As many of the other answer have pointed out, we've done astonishing things with limited computing power. The Space Shuttle AP-101s were roughly on par with the power of a Nintendo Game Boy! (Gameboy is faster but 8 bit, AP-101 is slower but could do 32 bit arithmetic). It also relied on core memory, which looks something like this:
(Yes, those are wires woven through individual ferrite cores. From what I have read, much of core memory was produced by women because it was hard to find men who could weave such wires with enough precision)
And this was a powerful computer. Gemini ran on a 7,000 instruction per second computer.
So we did accomplish a lot with old technologies. However, there's one common thread to all space exploration so far: there was no space combat.
In space combat, you're not just up against nature, you're up against another individual who is trying to kill you with the utmost ferocity. That changes the game somewhat. The Apollo missions may have time to have ground control calculate a course deviation and then punch it into a computer. If you have a salvo of weapons bearing in on you, you must respond much faster.
And herein lies the rub. Your technological civilization has one brutal handicap. This means that, unless the magical civilization has a similar handicap, no amount of "can it be done" will be sufficient. The magic users will simply out maneuver, out defend, out attack, and overall out-win your technology users. You might as well have asked if Bear Grylls can survive on a modern battlefield unarmed. Sure, his survival skills are famous, and he could probably actually find usable food and water on the battlefield, but the army of individuals with guns and tanks and aircraft is going to win.
Assuming you properly handicap the mages, this could work. The real secret to many of these devices is that we push them to the limits. We use computers to stabilize them, but that's only because we pushed it to begin with. If we relaxed our standards a bit, we'd find that things are much easier to manufacture. They won't be as good, but they'll work. And anyone knows that in combat, something working is better than something not working.
There will be limits. You wont have the same delta-V that you would with your computers. However, in many cases you can resolve this because you still have computers back home. Build the warships at the edges of your gravity wells, where escape velocity is as low as possible, and you don't need quite as much delta-V. Use your efficient craft where you can, and only send the dumb craft where you have to.
You will have to rely on dumb weapons, which will be more of a challenge. This will mean getting closer to the enemy than you are comfortable with, and wasting more reaction mass than you are comfortable with. If the mages are on their game, this will probably be a death wish, but if you've handicapped the mages properly, you may get away with it.
A big challenge will be sensing. The distances in realistic space combat are enormous. Without modern computers and ICs providing your sensing technologies, it's going to be hard to see your opponent before they whizz past you. Fortunately, the enemy will have the same limitations as you. Maybe you can borrow some of their magic amulets or what-not to sense with!